轴向加速运动黏弹性梁受迫振动中的混沌动力学

研究了黏弹性轴向运动梁在外部激励和参数激励共同作用下横向振动的混沌非线性动力学行为. 引入有限支撑刚度, 并考虑黏弹性本构关系取物质导数, 同时计入由梁轴向加速度引起的沿径向变化的轴力, 建立轴向运动黏弹性梁横向非线性振动的偏微分-积分模型. 通过Galerkin截断方法研究了外部激励的频率和因速度简谐脉动引起的参数激励的频率在不可通约关系时轴向运动连续体的非线性动力学行为, 并对不同截断阶数的数值预测进行了对比. 基于对控制方程的Galerkin截断, 得到离散化的常微分方程组, 使用四阶Runge-Kutta方法求解. 基于此数值解, 运用非线性动力学时间序列分析方法, 通过Poincaré 映射, 观察到轴向运动梁随扰动速度幅值的倍周期分岔现象, 并比较了有无外部激励对倍周期分岔的影响. 分别在低速以及近临界高速运动状态下, 从相平面图、Poincaré 映射以及频谱分析的角度识别了系统中存在的准周期运动形态. 关键词： 轴向运动梁 非线性 混沌 分岔     

Nonlinear free transverse vibration of an axially moving beam: Comparison of two models

Nonlinear free transverse vibration of an axially moving beam is investigated. A partial-differential equation governing the transverse vibration is derived from the Newton's second law. Under the assumption that the tension of beam can be replaced by the averaged tension over the beam, the partial-differential reduces to a widely used integro-partial-differential equation for nonlinear free transverse vibration. The method of multiple scales is applied directly to two equations to evaluate nonlinear natural frequencies. Numerical examples are presented to demonstrate the analytical results and to highlight the difference between two models. Two models yield the essentially same results for the weak nonlinearity, the small axial speed and the low mode, while the difference between two models increases with the nonlinear term, the axial speed, and the order of mode.     

Theoretical and experimental study on the transverse vibration properties of an axially moving nested cantilever beam

An axially moving nested cantilever beam is a type of time-varying nonlinear system that can be regarded as a cantilever stepped beam. The transverse vibration equation for the axially moving nested cantilever beam with a tip mass is derived by D’Alembert?s principle, and the modified Galerkin?s method is used to solve the partial differential equation. The theoretical model is modified by adjusting the theoretical beam length with the measured results of its first-order vibration frequencies under various beam lengths. It is determined that the length correction value of the second segment of the nested beam increases as the structural length increases, but the corresponding increase in the amplitude becomes smaller. The first-order decay coefficients are identified by the logarithmic decrement method, and the decay coefficient of the beam decreases with an increase in the cantilever length. The calculated responses of the modified model agree well with the experimental results, which verifies the correctness of the proposed calculation model and indicates the effectiveness of the methods of length correction and damping determination. Further studies on non-damping free vibration properties of the axially moving nested cantilever beam during extension and retraction are investigated in the present paper. Furthermore, the extension movement of the beam leads the vibration displacement to increase gradually, and the instantaneous vibration frequency and the vibration speed decrease constantly. Moreover, as the total mechanical energy becomes smaller, the extension movement of the nested beam remains stable. The characteristics for the retraction movement of the beam are the reverse.     

Galerkin method for steady-state response of nonlinear forced vibration of axially moving beams at supercritical speeds

The present paper investigates the steady-state periodic response of an axially moving viscoelastic beam in the supercritical speed range. The straight equilibrium configuration bifurcates in multiple equilibrium positions in the supercritical regime. It is assumed that the excitation of the forced vibration is spatially uniform and temporally harmonic. Under the quasi-static stretch assumption, a nonlinear integro-partial-differential equation governs the transverse motion of the axially moving beam. The equation is cast in the standard form of continuous gyroscopic systems via introducing a coordinate transform for non-trivial equilibrium configuration. For a beam constituted by the Kelvin model, the primary resonance is analyzed via the Galerkin method under the simply supported boundary conditions. Based on the Galerkin truncation, the finite difference schemes are developed to verify the results via the method of multiple scales. Numerical simulations demonstrate that the steady-state periodic responses exist in the transverse vibration and a resonance with a softening-type behavior occurs if the external load frequency approaches the linear natural frequency in the supercritical regime. The effects of the viscoelastic damping, external excitation amplitude, and nonlinearity on the steady-state response amplitude for the first mode are illustrated.     

Stabilization of an axially moving web via regulation of axial velocity

In this paper, a novel control algorithm for suppression of the transverse vibration of an axially moving web system is presented. The principle of the proposed control algorithm is the regulation of the axial transport velocity of an axially moving beam so as to track a profile according to which the vibration energy decays most quickly. The optimal control problem that generates the proposed profile of the axial transport velocity is solved by the conjugate gradient method. The Galerkin method is applied in order to reduce the partial differential equation describing the dynamics of the axially moving beam into a set of ordinary differential equations (ODEs). For control design purposes, these ODEs are rewritten into state-space equations. The vibration energy of the axially moving beam is represented by the quadratic form of the state variables. In the optimal control problem, the cost function modified from the vibration energy function is subjected to the constraints on the state variables, and the axial transport velocity is considered as a control input. Numerical simulations are performed to confirm the effectiveness of the proposed control algorithm.     

Nonlinear dynamics of higher-dimensional system for an axially accelerating viscoelastic beam with in-plane and out-of-plane vibrations

In this paper, the bifurcations and chaotic motions of higher-dimensional nonlinear systems are investigated for the nonplanar nonlinear vibrations of an axially accelerating moving viscoelastic beam. The Kelvin viscoelastic model is chosen to describe the viscoelastic property of the beam material. Firstly, the nonlinear governing equations of nonplanar motion for an axially accelerating moving viscoelastic beam are established by using the generalized Hamilton’s principle for the first time. Then, based on the Galerkin’s discretization, the governing equations of nonplanar motion are simplified to a six-degree-of-freedom nonlinear system and a three-degree-of-freedom nonlinear system with parametric excitation, respectively. At last, numerical simulations, including the Poincare map, phase portrait and Lyapunov exponents are used to analyze the complex nonlinear dynamic behaviors of the axially accelerating moving viscoelastic beam. The bifurcation diagrams for the in-plane and out-of-plane displacements via the mean axial velocity, the amplitude of velocity fluctuation and the frequency of velocity fluctuation are respectively presented when other parameters are fixed. The Lyapunov exponents are calculated to identify the existence of the chaotic motions. From the numerical results, it is indicated that the periodic, quasi-periodic and chaotic motions occur for the nonplanar nonlinear vibrations of the axially accelerating moving viscoelastic beam. Observing the in-plane nonlinear vibrations of the axially accelerating moving viscoelastic beam from the numerical results, it is found that the nonlinear responses of the six-degree-of-freedom nonlinear system are much different from that of the three-degree-of-freedom nonlinear system when all parameters are same.     

Analysis of the coupled thermoelastic vibration for axially moving beam

The coupled thermoelstic vibration characteristics of the axially moving beam are investigated. The differential equation of motion of the axially moving beam under the thermoelastic coupling is established based to the equilibrium equation and the thermal conduction equation involving deformation term. The eigenequation is deduced and the dimensionless complex frequencies of the axially moving beam with different boundary conditions under the coupled thermoelastic case are calculated by the differential quadrature method. The curves of the real parts and imaginary parts of the first three-order dimensionless complex frequencies versus the dimensionless axially moving speed are obtained. The effects of the dimensionless coupled thermoelastic factor, the ratio of length to height, the dimensionless moving speed on the stability of the beam are analyzed.     

Stability and bifurcation of an axially moving beam tuned to three-to-one internal resonances

This study analyzed the nonlinear vibration of an axially moving beam subject to periodic lateral force excitations. Attention is paid to the fundamental and subharmonic resonances, since the excitation frequency is close to the first two natural frequencies of the system. The incremental harmonic balance (IHB) method was used to evaluate the nonlinear dynamic behaviour of the axially moving beam. The stability and bifurcations of the periodic solutions for given parameters were determined by the multivariable Floquet theory using Hsu’s method. The solutions obtained from the IHB method agreed very well with those obtained from numerical integration. Furthermore, numerical examples are given to illustrate the effects of the three-to-one internal resonance on the response of the system.     

Galerkin methods for natural frequencies of high-speed axially moving beams

In this paper, natural frequencies of planar vibration of axially moving beams are numerically investigated in the supercritical ranges. In the supercritical transport speed regime, the straight equilibrium configuration becomes unstable and bifurcate in multiple equilibrium positions. The governing equations of coupled planar is reduced to two nonlinear models of transverse vibration. For motion about each bifurcated solution, those nonlinear equations are cast in the standard form of continuous gyroscopic systems by introducing a coordinate transform. The natural frequencies are investigated for the beams via the Galerkin method to truncate the corresponding governing equations without nonlinear parts into an infinite set of ordinary-differential equations under the simple support boundary. Numerical results indicate that the nonlinear coefficient has little effects on the natural frequency, and the three models predict qualitatively the same tendencies of the natural frequencies with the changing parameters and the integro-partial-differential equation yields results quantitatively closer to those of the coupled equations.     

Dynamic analysis of an axially moving finite-length beam with intermediate spring supports

This paper presents the analysis for the transverse vibration of an axially moving finite-length beam inside which two points are supported by rotating rollers. In this study, the rollers are modeled as uniaxial springs in the transverse direction. Hamilton?s principle is applied to derive the equations of motion and boundary conditions of the system. The equations of motion include translational and rotational motions as well as flexible motion. These equations are discretized using Galerkin?s method, and then the dynamic characteristics of a flexible beam with spring supports are studied by solving an eigenvalue problem. The veering phenomenon of natural frequency loci and mode exchanges are investigated for different positions of the springs and various values of the spring stiffness. In addition, the mode localization is also analyzed using the peak amplitude ratio. It is found in this study that the first mode is localized in one of the beam spans if an appropriate value of the spring constant is selected. Furthermore, it is shown that mode localization can be used to reduce the vibration transferred from one span to the other span while a beam moves axially.     

Free vibration and stability of a cantilever beam attached to an axially moving base immersed in fluid

Free vibration and stability are investigated for a cantilever beam attached to an axially moving base in fluid. The equations of motion of the slender cantilever beam affiliated to an axially moving base at a known rate while immersed in an incompressible fluid are derived first. An “axially added mass coefficient” is taken into account in the obtained equations. Then, a coordinate transformation is introduced to fix the boundaries. Based on the Galerkin approach, the natural frequencies of the beam system are numerically analyzed. The effects of moving speed of the base and several other system parameters on the dynamics and stability of the beam are discussed in detail. It is found that when the moving speed exceeds a certain value the beam becomes unstable and the instability type is sensitive to the system parameters. When the values of system parameters, such as mass ratio and axially added mass coefficient, are big enough, however, no instabilities are detected. The variations of the lowest unstable critical moving speed with respect to several key parameters are also investigated.     

Nonlinear dynamic behaviors of a deploying-and-retreating wing with varying velocity

In this paper, the nonlinear dynamical behaviors of deploying-and-retreating wings in supersonic airflow are investigated. A cantilever laminated composite beam, which is axially moving at a known rate, is implemented to model the deploying-and-retreating wing. Associated with Reddy's third-order theory and von Karman type equations of large deformation, the nonlinear governing equations of motion of the deploying-and-retreating wing are derived based on the Hamilton's principle. The nonlinear partial differential equations of motion are transformed into a set of the ordinary differential equations using Galerkin's method. The nonlinear dynamical behaviors of the deployable-and-retreating wing are investigated in the cases of three different axially moving rates during deploying process and retreating process using the numerical simulations.     

Longitudinal,transverse, and torsional vibrations and stabilities of axially moving single-walled carbon nanotubes

Thanks to the brilliant mechanical properties of single-walled carbon nanotubes (SWCNTs), they are suggested as high speed nanoscale vehicles. To date, various aspects of vibrations of SWCNTs have been addressed; however, vibrations and instabilities of moving SWCNTs have not been thoroughly assessed. Herein, vibrational properties of an axially moving SWCNT with simply supported ends are studied using nonlocal Rayleigh beam theory. Employing assumed mode and Galerkin methods, the discrete governing equations pertinent to longitudinal, transverse, and torsional motions of the moving SWCNT are obtained. The resulting eigenvalue equations are then numerically solved. The speeds corresponding to the initiation of the instability within the moving nanostructure are calculated. The roles of the speed of the moving SWCNT, small-scale parameter, and aspect ratio on the characteristics of longitudinal, transverse, and torsional vibrations of axially moving SWCNTs are scrutinized. The obtained results show that the appearance of the small-scale parameter would result in the occurrence of both divergence and flutter instabilities at lower levels of the speed.     

Parabolic equation for nonlinear acoustic wave propagation in inhomogeneous moving media

A new parabolic equation is derived to describe the propagation of nonlinear sound waves in inhomogeneous moving media. The equation accounts for diffraction, nonlinearity, absorption, scalar inhomogeneities (density and sound speed), and vectorial inhomogeneities (flow). A numerical algorithm employed earlier to solve the KZK equation is adapted to this more general case. A two-dimensional version of the algorithm is used to investigate the propagation of nonlinear periodic waves in media with random inhomogeneities. For the case of scalar inhomogeneities, including the case of a flow parallel to the wave propagation direction, a complex acoustic field structure with multiple caustics is obtained. Inclusion of the transverse component of vectorial random inhomogeneities has little effect on the acoustic field. However, when a uniform transverse flow is present, the field structure is shifted without changing its morphology. The impact of nonlinearity is twofold: it produces strong shock waves in focal regions, while, outside the caustics, it produces higher harmonics without any shocks. When the intensity is averaged across the beam propagating through a random medium, it evolves similarly to the intensity of a plane nonlinear wave, indicating that the transverse redistribution of acoustic energy gives no considerable contribution to nonlinear absorption. Published in Russian in Akusticheskiĭ Zhurnal, 2006, Vol. 52, No. 6, pp. 725–735. This article was translated by the authors.     

Complex geometrical optics of Kerr type nonlinear media

The paper generalizes paraxial complex geometrical optics (PCGO) for Gaussian beam (GB) propagation in nonlinear media of Kerr type. Ordinary differential equations for the beam amplitude and for complex curvature of the wave front are derived, which describe the evolution of axially symmetric GB in a Kerr type nonlinear medium. It is shown that PCGO readily provides the solutions of NLS equation obtained earlier from diffraction theory on the basis of the aberration-free approach. Besides reproducing classical results of self-focusing PCGO readily describes an influence of the initial curvature of the wave front on the beam evolution in a medium of Kerr type including a nonlinear graded-index fiber. The range of applicability of the PCGO theory is discussed as well which is helpful for avoiding nonphysical solutions.     

Simultaneous control of longitudinal and transverse vibrations of an axially moving string with velocity tracking

In this paper, an active control scheme for an axially moving string system that suppresses both longitudinal and transverse vibrations and regulates the transport velocity of the string to track a desired moving velocity profile is investigated. The control scheme utilizes three inputs: one control force at the right boundary, which is exerted by a hydraulic actuator equipped with a damper, and two control torques applied at the left and right rollers. The equations of motion are derived by using Hamilton's principle. Two nonlinear partial differential equations govern the longitudinal and transverse motions, where the variation of the tension of the string due to the transverse and longitudinal vibrations is considered. Among four boundary conditions, two describe the rotational dynamics of the left and right rollers; one determines the dynamics of the hydraulic actuator at the right boundary, and the last one denotes that the left boundary is fixed. The Lyapunov method is employed to generate control laws. Asymptotic stability of the transverse and longitudinal dynamics and the velocity tracking error is achieved. The effectiveness of the proposed control scheme is illustrated via numerical simulations.     

Theory of a rotating-mode coaxial infrared grating laser

Analysis is presented for a novel interaction that makes possible a compact infrared grating laser configuration. The laser utilizes an annular axis encircling beam that is confined by an axial magnetic field and passes along a grating blazed axially on the center conductor of a coaxial resonator. A smooth cylindrical outer conductor completes the Fabry-Perot resonator to provide the necessary feedback for sustained oscillation. Linear analysis leads to an eigenvalue equation for cavity resonance frequencies, start-oscillation currents, and a gain-bandwidth relationship. The latter permits estimation of an upper bound on the ideal-beam power extraction efficiency. The nonlinear studies make use of analytical and numerical methods to gain a detailed understanding of electron motion in the presence of both axially focusing and radio frequency fields, and to determine the nonlinear efficiency. There is significant degradation of the interaction efficiency when the beam has finite annular thickness. Efficiency enhancement by means of a down-tapered axial field is demonstrated. A point design for lasing at ~160 μm in the far-infrared, utilizing a 100-kV beam with a pitch angle of 55° is presented. The start-oscillation current is a sensitive function of the resonator quality factor (or finesse); for a well-designed resonator, it is on the order of tens of Amperes     

Free torsional vibration of nanotubes based on nonlocal stress theory

A new elastic nonlocal stress model and analytical solutions are developed for torsional dynamic behaviors of circular nanorods/nanotubes. Unlike the previous approaches which directly substitute the nonlocal stress into the equations of motion, this new model begins with the derivation of strain energy using the nonlocal stress and by considering the nonlinear history of straining. The variational principle is applied to derive an infinite-order differential nonlocal equation of motion and the corresponding higher-order boundary conditions which contain a nonlocal nanoscale parameter. Subsequently, free torsional vibration of nanorods/nanotubes and axially moving nanorods/nanotubes are investigated in detail. Unlike the previous conclusions of reduced vibration frequency, the solutions indicate that natural frequency for free torsional vibration increases with increasing nonlocal nanoscale. Furthermore, the critical speed for torsional vibration of axially moving nanorods/nanotubes is derived and it is concluded that this critical speed is significantly influenced by the nonlocal nanoscale.     

Waveguide propagation of sound beams in a nonlinear medium

A possibility of a waveguide propagation of sound beams in the case of compensation of the diffraction divergence by the nonlinear refraction is demonstrated theoretically. A stationary (with respect to the longitudinal coordinate) solution is obtained to the nonlinear equation for a sound beam (the Khokhlov—Zabolotskaya equation); the solution describes the characteristic bow-shaped profile of the beam and the self-localized (with respect to the transverse coordinate) distribution of the peak values of this profile. The physical and mathematical features of this phenomenon belonging to nonlinear acoustics are discussed and compared with those of the well-known analog from nonlinear optics. A scheme of an experimental realization of the waveguide propagation of acoustic beams is proposed.     

Steady-state responses of axially accelerating viscoelastic beams: Approximate analysis and numerical confirmation

Nonlinear parametric vibration of axially accelerating viscoelastic beams is investigated via an approximate analytical method with numerical confirmations. Based on nonlinear models of a finite-small-stretching slender beam moving at a speed with a periodic fluctuation, a solvability condition is established via the method of multiple scales for subharmonic resonance. Therefore, the amplitudes of steady-state periodic responses and their existence conditions are derived. The amplitudes of stable steady-state responses increase with the amplitude of the axial speed fluctuation, and decrease with the viscosity coefficient and the nonlinear coefficient. The minimum of the detuning parameter which causes the existence of a stable steady-state periodic response decreases with the amplitude of the axial speed fluctuation, and increases with the viscosity coefficient. Numerical solutions are sought via the finite difference scheme for a nonlinear partial-differential equation and a nonlinear integro-partial-differential equation. The calculation results qualitatively confirm the effects of the related parameters predicted by the approximate analysis on the amplitude and the existence condition of the stable steady-state periodic responses. Quantitative comparisons demonstrate that the approximate analysis results have rather high precision. Supported by the National Outstanding Young Scientists Foundation of China (Grant No. 10725209), the National Natural Science Foundation of China (Grant No. 10672092), Scientific Research Project of Shanghai Municipal Education Commission (Grant No. 07ZZ07), and Shanghai Leading Academic Discipline Project (Grant No. Y0103)